The researchers say that even mammals that breed year round—which should offer protection against seasonal shifts—may still feel the impact of climate change.

I don't understand what the word "which" means here. Does it refer to the "mammals"? If it does, I am confused by the whole sentence — what's the connection between the middle part of the sentence and the rest?

It refers to [the fact] that they breed year round. The word which can be used to refer to an entire sentence, to a specific noun phrase, to any other part of a sentence, or even to a concept that isn't explicitly mentioned. Several other pronouns can do the same:

Christians refused to worship the Emperor. That was not to the liking of the Roman government.